Investigation on optimal spray properties for groundbased agricultural applications using deposition and

retention models

Nicolas De Cocka,b,∗, Mathieu Massinona, Sofiene Ouled Taleb Salaha,c,d,Frederic Lebeaua

aTERRA Teaching and Research Centre, Gembloux Agro-Bio Tech, University of Liege,B-5030 Gembloux, Belgium

bvon Karman Institute for Fluid Dynamics, B-1640 Rhode-Saint-Genese, BelgiumcCESAM - GRASP, Institute of Physics, University of Liege, Building B5a, Sart Tilman,

B-4000 Liege, BelgiumdTERRA - AgricultureIsLife, Gembloux Agro-Bio Tech, Passage des deportes 2, University

of Liege, B-5030 Gembloux, Belgium

Abstract

In crop protection, it is well know that droplet size determine spray efficacy.

The optimisation of both spray deposition and retention leads to a dilemma:

should small droplets be used to increase retention or large droplets be pre-

ferred to avoid drift? An ideal droplet should have a short time of flight to

minimise its distance travelled while impacting the target with a moderate ki-

netic energy. This paper aims to determine an optimum range of droplet sizes

for boom-sprayer applying herbicide using a modelling approach. The main

parameters of spray deposition and retention models are systematically varied

and the effects on drift potential and droplet impaction outcomes are discussed.

The results of the numerical simulations showed that droplets with diameter

ranging between 200µm and 250µm offer high control of deposition by combin-

ing a low drift potential and a moderate kinetic energy at top of the canopy. A

fourfold reduction of the volume drifting further than 2 m from the nozzle was

observed for a spray with a volume median diameter of 225µm when the relative

span factor of the droplet spectrum was reduced from 1.0 to 0.6. In the latter

∗Corresponding authorEmail address: nicolas.decock@ulg.ac.be (Nicolas De Cock)

Preprint submitted to Journal of LATEX Templates August 3, 2017

scenario, an increase from 63 to 67% of the volumetric proportion of droplets

adhering to the wheat leaf was observed. Therefore, strategies for controlling

the droplet size distribution may offer promising solutions for reducing adverse

impact of spray applications on environment.

Keywords: stochastic Lagrangian; agricultural spray; droplet size

distribution; drift; retention; relative span factor; deposition; retention;

1. Introduction1

Spray application is a key process in crop protection to ensure high yields2

whilst minimising the adverse environmental and health impact of plant protec-3

tion products. During this process, the agricultural mixture is usually atomised4

by passage through a nozzle generating a liquid sheet that further breaks up5

in a cloud of droplets. A herbicide application can be divided in four succes-6

sive stages: deposition (initial spray amount minus off-target losses), retention7

(amount remaining on the plant after impaction), uptake (amount of active in-8

gredient taken into the plant foliage) and translocation (amount of absorbed9

material translocated) (Zabkiewicz, 2007). This paper focuses on deposition10

and retention stages.11

It has been shown that the droplet size distribution of the spray significantly12

affects the deposition (Hilz & Vermeer, 2013; Nuyttens, De Schampheleire,13

Baetens & Sonck, 2007b; Stainier, Destain, Schiffers & Lebeau, 2006; Taylor,14

Womac, Miller & Taylor, 2004).Al Heidary, Douzals, Sinfort & Vallet (2014)15

showed that spray drift decreases with the droplet kinetic energy following a16

power law. Indeed, finer droplets are more prone to drift leading to poten-17

tial product losses in the air, water and soil (Reichenberger, Bach, Skitschak18

& Frede, 2007). Modelling of deposition under field conditions has been real-19

ized using several approaches: Gaussian plume model (Baetens, Ho, Nuyttens,20

De Schampheleire, Melese Endalew, Hertog, Nicolai, Ramon & Verboven, 2009;21

Lebeau, Verstraete, Stainier & Destain, 2011; Raupach, Briggs, Ford, Leys,22

2

Nomenclature

Greek Symbols

β Droplet release angle [◦]

∆t Time step [s]

η, ε Random value from a standard

normal distribution [-]

γ Surface tension [Nm−1]

κ von Karman constant [-]

λ,K Weibull distribution parameter [-]

µ Dynamic viscosity [N s m−2]

ν Kinematic viscosity [m s−2]

ρ Volumetric mass [kg m−3]

σx,z Velocity RMS [m s−1]

τL Lagrangian time scale of turbu-

lence [s]

τ∗L Modified Lagrangian timescale [s]

θ Static contact angle [◦]

Roman Symbols

m Mass flux [kg s−1]

CD Drag coefficient [-]

CDF Cumulative density function [-]

d Droplet diameter [m]

d0 Zero plane displacement [m]

dm Maximum spread diameter [m]

E Arithmetic mean of droplet trav-

eled distance [m]

g Gravity acceleration [m s−2]

hc Crop height [m]

hr Release height [m]

k Liquid to gas dynamic viscosity

ratio [-]

L Monin-Obukhov length [m]

m Droplet mass [kg]

Re Reynolds number [-]

RSF Relative span factor [-]

ToF Time of flight [s]

U Air flow velocity [m s−1]

u Droplet velocity [m s−1]

U∗ Friction velocity [m s−1]

u0 Release velocity [m s−1]

Vr Relative droplet velocity [m s−1]

We Weber number [-]

x Horizontal position [m]

z Vertical position [m]

z0 Surface roughness [m]

Subscripts

g Gaseous

l Liquid

Muschal, Cooper & Edge, 2001), Lagrangian models (Butler Ellis & Miller,23

2010; Holterman, Van De Zande, Porskamp & Huijsmans, 1997; Mokeba, Salt,24

Lee & Ford, 1997; Teske, Bird, Esterly, Curbishley, Ray & Perry, 2002; Walk-25

late, 1987), computational fluid dynamics (CFD) (Baetens, Nuyttens, Verboven,26

De Schampheleire, Nicolai & Ramon, 2007; Weiner & Parkin, 1993). Here a La-27

grangian stochastic model will be used. Lagrangian stochastic models compute28

3

the droplet movement through an airflow using discrete time steps. The airflow29

turbulence is taken into account by superposing a time correlated fluctuating30

component onto a mean component. Dispersal statistics can be retrieved by31

tracking a large number of droplets.32

The amount of spray remaining on a plant after impact is determined by the33

sum of each droplet impact outcomes (adhesion, bounce or shatter). Droplet34

behaviour after impact is mainly governed by droplet kinetic energy, liquid sur-35

face tension and the surface wetability (Josserand & Thoroddsen, 2016; Yarin,36

2006). When a droplet hits a solid surface, it spreads radially producing a thin37

liquid layer. If the droplet kinetic energy at impact overcomes capillary forces,38

the droplet shatters in smaller droplets. Otherwise, the spreading driven by the39

initial kinetic energy of the droplet is decelerated by viscous forces and surface40

tension, until radial dispersion stops. Thereafter, the liquid layer can remain41

pinned on the surface or retract. If the droplet surface energy is sufficient, the42

droplet may detach itself from the surface leading to a bounce (Attane, Girard43

& Morin, 2007). Otherwise, the droplet adheres on the surface. Massinon,44

Dumont, De Cock, Salah & Lebeau (2015) proposed an empirical probabilistic45

model using droplet Weber number to model droplet outcomes on plant leaves.46

Deterministic models of impact outcomes based on energy balance of the impact-47

ing droplet are also available (Mao, Kuhn & Tran, 1997; Mundo, Sommerfeld48

& Tropea, 1995; Dorr, Wang, Mayo, McCue, Forster, Hanan & He, 2015).49

One common approach to reduce drift is to shift the droplet spectrum to-50

wards coarser droplets using low-drift nozzle or by adding spray additives. How-51

ever, coarse droplets present a relatively low degree of surface coverage and may52

bounce or shatter on the target (Hilz & Vermeer, 2013; Massinon, De Cock,53

Forster, Nairn, McCue, Zabkiewicz & Lebeau, 2017). An other solution, is to54

narrow the droplet size distribution towards an intermediate range of droplet55

size.56

The goal of the present paper is to determine an optimum range of droplet57

size for boom-sprayer based herbicide applications using a modelling approach.58

A deposition model based on a stochastic Lagrangian approach is presented in59

4

the section 2.1. The mathematical models determining the droplet outcomes at60

canopy level are presented in the section 2.2. Deposition and retention models61

are used to realise a sensitivity analysis on initial droplet parameters (diameter,62

release height, release velocity) and environmental characteristics (wind speed,63

relative humidity) in the agricultural range detailed in section 2.4.1. Finally, the64

aerial transport and the retention of sprays with different volumetric median65

diameter and relative span factor are assessed in section 2.4.2.66

2. Materials and methods67

2.1. Droplet deposition model68

2.1.1. General overview of the droplet transport model69

Figure 1 shows the flow chart of the model. The simulation starts by ini-70

tialising the droplet characteristics, e.g. its initial location, velocity and size.71

The acceleration and the evaporation of the droplet is then computed at each72

time step. In order to solve the aerodynamic balance of the droplet, the air73

flow characteristics are computed as well at each droplet location taking into74

account atmospheric turbulence. The simulation ends when the droplet either75

looses all its mass or reaches the crop level canopy where the droplet is stated76

to be captured. Air entrainment from the spray nozzle is not taken in account77

in the present model because of a low drop/air mass ratio is assumued which is78

typical of low application volume/high speed applications (Lebeau, 2004).79

2.1.2. Droplet motion80

Equations of droplet motion are taken from the saltation model of Kok81

& Renno (2009), which takes into account the particle inertia. The droplet82

transport model uses a Lagrangian description of the droplet motion. The83

displacement of the droplet after a time t is given by the numerical integration84

of the droplet velocity over time:85

∆xi =

n∑i

ux,i ∆ti & ∆zi =

n∑i

uz,i ∆ti (1)

5

∂mdt

Eq 6

∂udt

Eq 9Eq 2

Uz,iUx,i

Fig. 1: Flow chart of the droplet transport model.

The variation of the droplet velocity is retrieved using Newton’s second law of86

motion taking in account the effects of drag and gravity while neglecting the87

buoyancy.88

∂uxdt =

3CD ρgVr(Ux−ux)4 ρl d

∂uzdt =

3CD ρgVr(Uz−uz)4 ρl d

− g(2)

with m the droplet mass [kg], u the droplet velocity [m s−1], t the time [s], CD89

the drag coefficient [-], ρl and ρg the density of the liquid and gaseous phases90

respectively [kg m−3], A the droplet cross section area [m2], d the droplet diam-91

eter [m], U and u are the air and the droplet velocity respectively [m s−1], Vr is92

the relative velocity between the droplet and the airflow defined as Vr =| u− U |93

[m s−1], and g is the gravitational acceleration rounded to 9.81 [m s−2].94

For Re≤ 400 the drag coefficient of a sphere in a gas flow can be expressed95

using the following expression (Saboni, Alexandrova & Gourdon, 2004):96

Cd =

(k(24Re + 4

Re0.36

)+ 15

Re0.82 − 0.02kRe0.5

1+k

)Re2 + 40 3k+2

Re + 15k + 10

(1 + k)(5 + 0.95Re2)(3)

6

with k equal to the ratio of the liquid to the gas viscosity, k = µlµg

, and Re97

the droplet Reynolds number defined as: Re = Vrdνg

. Other Cd expressions for98

a sphere can be found in the literature (Barati, Neyshabouri & Ahmadi, 2014;99

Langmuir & Blodgett, 1949).100

2.1.3. Description of the air flow101

The velocity profile generated by a wind above crop is made up of a random102

part sum onto a mean component. Assuming the vertical mean flow equal to103

zero, the general formulation is reduced to:104

Ux = Ux + U ′x ; Uz = U ′z (4)

The average part of the horizontal velocity U(zi) is described by a logarith-105

mic velocity profile:106

U(zi) =U∗

κlog

(z − d0z0

)(5)

with κ the von Karman constant equals to 0.41 [-], U∗ the friction velocity107

[m s−1], z the distance above the ground [m], d0 the zero plane displacement108

[m] and z0 the surface roughness [m]. The values d0 and z0 can be related to crop109

height using z0 = 0.1hc and d0 = 0.63hc with hc the crop height (Butler Ellis110

& Miller, 2010).111

For homogeneous isotropic turbulence, the velocity fluctuations U ′ of an air112

particle moving with the flow can be statically described by the following set of113

equations (Kok & Renno, 2009) (Wilson & Sawford, 1996):114

U ′x,i+1 = U ′x,i e− ∆tτ∗Lx + ε σx

√2

(1− e

(−√

∆tτ∗Lx

))

U ′z,i+1 = U ′z,i e− ∆tτ∗Lz + η σz

√2

(1− e

(−√

∆tτ∗Lz

)) (6)

with τ∗L the modified Lagrangian time scale [s], ∆t is the time step [s], η and115

ε are random variables from a standard normal distribution [-], σx and σz are116

the horizontal and the vertical velocity fluctuations [m s−1]. For near neutral117

7

atmospheric conditions: σx = 2.3U∗ and σz = 1.3U∗ (Panofsky, Tennekes,118

Lenschow & Wyngaard, 1977).119

The Lagrangian timescale represents the approximate timescale over which120

the velocities experienced by an air particle are statically related. Since the121

droplets move through the air eddies, the Lagrangian timescale perceived by122

the droplets is shorter. A modified formulation of the Lagrangian timescale for123

the horizontal and the vertical directions was proposed by (Sawford & Guest,124

1991):125

τ∗Lx = τLx√1+(2 Vrσx )

2

τ∗Lz = τLz√1+(Vrσz )

2

(7)

with τL defined as (Butler Ellis & Miller, 2010):126

τL = κU∗(z − d0)

σ2z

√1−

(16(z − d0)

L

)(8)

with L the Monin-Obukhov length [m], which characterises atmospheric stabil-127

ity.128

2.1.4. Droplet evaporation129

Droplet evaporation in the model was based on Guella, Alexandrova &130

Saboni (2008). The set of equations used are described in the Appendix A.131

In this model, the air has a constant vapour fraction and temperature. The loss132

of droplet volume is computed after each time step as:133

∆m =m

ρl∆t (9)

134

2.2. Droplet retention model135

Mathematical models have been developed to predict the outcome of impact-

ing droplets based on an energy balance approach(Dorr et al., 2015; Mao et al.,

8

1997; Mundo et al., 1995). In these models, three impact outcomes are consid-

ered: adhesion, bounce or shatter. Shatter occurs when the inertial forces at

impacting overcome the capillary forces. The droplet shatter threshold may be

predicted based on droplet Reynolds number and Weber number Mundo et al.

(1995):

K = We0.5Re0.25I (10)

Unlike for the drag coefficient, the Reynolds number of the droplet at im-

paction ReI is computed using the liquid kinematic viscosity: ReI = uzdνl

.

The Weber number is expressed as: We =u2z ρl dγ with γ the liquid tension

surface [N m−1]. Experimental measurements have shown that the droplets

shatter when We0.5Re0.25I ≥ Kcrit (Mundo et al., 1995). If the droplet does

not shatter, the model assesses the bounce criteria. Mao et al. (1997) proposed

a semi-empirical model based on energy conservation providing a rebound crite-

ria. Bounce occurs if the excess rebound energy E∗ERE is positive otherwise the

droplet is predicted to adhere to the leaf. The excess rebound energy is defined

as:

E∗ERE =1

4

(dmd

)2

(1−cos θ)+2

3

(d

dm

)−0.12

(dmd

)2.3

(1−cos θ)0.63−1 (11)

with dm the maximum spread diameter [m] and θ the static contact angle [◦].136

The value of dm in the Eq. 11 was, in turn, derived as an implicit function

of We, Re and θ:[0.25 (1− cos θ) + 0.2

We0.83

Re0.33I

](dmd

)3

−(We

12+ 1

)(dmd

)+

2

3= 0 (12)

If there is no real solution for dm in the Eq. 12 or if the computed dm is ≤ d,137

the value of dm is set as equal to d.138

2.3. Numerical procedure139

Figure 2 illustrates the initial state of the simulation. The initial droplet140

location is set as x = 0 and z = hr + hc with zr the release droplet height [m].141

The initial droplet velocity in both directions are: ux = ‖u0‖ cos(β) ; uz =142

‖u0‖ sin(β), with β the angle between the initial droplet direction and the143

9

d

β

hc

hr

z

x

Ux u0

Fig. 2: Initial configuration of the deposition model.

Table 1: Simulation constants. The air and water temperature properties were taken both for

15 ◦C. Subscript g and l refer to gaseous and liquid phases respectively.

Parameter Value Units

µg 1.85e-5 Pa s

µl 1.15e-3 Pa s

ρg 1.2 kg m−3

ρl 1000 kg m−3

hc 0.1 m

L -1000 m

vertical direction [◦] and u0 the release velocity [m s−1]. Liquid and air properties144

used for the computations are shown in the Table 1. The time step ∆t was145

computed as: min(

0.1hVr

, τL10

)[s].146

2.4. Parameter sensitivity study147

2.4.1. Monosized droplets148

A sensitivity analysis was performed to highlight the effect of the droplet di-149

ameter d, wind speed at a height of 2 m U(2), droplet release velocity u0, release150

angle β, the release height above crop hr and relative humidity Hr may have151

on the deposition and retention steps. The variation of these parameters are152

shown in the Table 2. For each instance, the trajectories of 15 000 droplets with153

the same initial conditions were computed. Random wind fluctuations experi-154

10

Table 2: Range of variation of the simulation parameters. The standard values are highlighted

in bold.

Variable Tested values Units

d 100;125;150;175;200;250;300;350;400 µm

U(2) 0;2;4;6;8 m s−1

‖u0‖ 5;10;15 m s−1

β 0;15;30;45;60;75;90 ◦

hr 0.25;0.5;0.75;1 m

Hr 40;60;80 %

enced by the droplets during their flights lead to a variety of trajectories that155

were characterised by statistical parameters such as mean, 5th, 50th (median)156

and 95th percentiles. Later in the paper, if the value of one parameter is not157

specified, the standard values indicated in bold in Table 2 were used.158

The impact outcomes were evaluated on a wheat leaf with water which has a159

static contact angle of 132 ◦ and a Kcrit of 69 (Forster, Mercer & Schou, 2010).160

Water has a surface tension γ of 0.072 N m−1.161

2.4.2. Polydisperse sprays162

The aerial transport of polydisperse sprays of droplets are simulated in order163

to predict the effect of the droplet size distribution on the overall deposition and164

retention. Each spray cloud was simulated by 100 000 droplets randomly drawn165

from a Weibull distribution in volumetric cumulative distribution (CDF) defined166

as: CDF = 1 − e−(−xλ )

K

(Rosin & Rammler, 1933; Babinsky & Sojka, 2002).167

The two Weibull distribution parameters were set to achieve a specific relative168

span factor RSF and volumetric mean diameter Dv50. The relative span factor169

is defined as RSF = Dv90−Dv10Dv50

with Dv10, Dv50 and Dv90 corresponding to the170

maximum droplet diameter below which 10 %, 50 % and 90 % of the volume of171

the sample exists, respectively. Six different values of Dv50 (150, 200, 225, 250,172

300, 350µm) and two RSF (0.6, 1) were simulated resulting in twelve different173

simulations. The twelve simulated droplet size distributions are shown in Fig.3.174

11

0 100 200 300 400 500 600

Diameter [µm]

0

10

20

30

40

50

60

70

80

90

100

Cu

mula

tive v

olu

me

pro

port

ion [

%]

Span = 0.6 VMD = 150 µm

Span = 0.6 VMD = 200 µm

Span = 0.6 VMD = 225 µm

Span = 0.6 VMD = 250 µm

Span = 0.6 VMD = 300 µm

Span = 0.6 VMD = 350 µm

Span = 1.0 VMD = 150 µm

Span = 1.0 VMD = 200 µm

Span = 1.0 VMD = 225 µm

Span = 1.0 VMD = 250 µm

Span = 1.0 VMD = 300 µm

Span = 1.0 VMD = 350 µm

Fig. 3: Cumulative droplet size distribution of the virtual sprays for the six Dv50 and the two

RSF .

Sprays characterised with a RSF of 0.6 and 1 are representative of the nar-175

row spray droplet size distributions produced by rotary atomisers (Qi, Miller176

& Fu, 2008) and flat fan nozzles respectively (De Cock, Massinon, Nuyttens,177

Dekeyser & Lebeau, 2016; Nuyttens, Baetens, De Schampheleire & Sonck, 2007a).178

A Dv50 of 250µm with a RSF of 1 is similar to a spray generated by a flat fan179

nozzle 110-03 operating at at 300 kPa. For all these cases, simulation parameters180

were set to standard values (Table 2).181

3. Results182

3.1. Sensitivity analysis of a population of monodisperse droplets183

3.1.1. Effect of droplet size on velocity dynamics184

Figure 4 a shows the evolution of the vertical median droplet velocity with185

respect to the droplet vertical position. The droplets are released 0.6 m above a186

crop of 0.1 m high with an initial horizontal velocity ux of 0 m s−1 and an initial187

vertical velocity uz of -10 m s−1. The droplets were decelerating in the vertical188

direction approaching their settling velocity whilst, in the horizontal direction,189

the droplets were accelerating towards the wind velocity. The droplets with a190

diameter≥ 250µm reached the crop canopy with a vertical velocity above their191

settling velocity. The median time of flight (ToF) for each droplet size is shown192

12

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Vertical position [m]

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

uz [

m s

-1]

1.7

9 s

0.70 s 0.31 s 0.16 s 0.10 s

0.08 s

100 µm

150 µm

200 µm

250 µm

300 µm

400 µm

a)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Vertical position [m]

0

0.5

1

1.5

ux [

m s

-1]

100 µm

150 µm

200 µm

250 µm

300 µm

400 µm

Wind profile

b)

Fig. 4: a) Median vertical velocity with respect to the droplet vertical location. The median

time of flight to travel from the release point to the crop top canopy for each droplet size is

indicated above each corresponding line. b) Median horizontal velocity with respect to the

droplet vertical location. The average wind velocity profile defined by the Eq.5 for a reference

wind of 2 m s−1 at 2 m is illustrated by the black curve.

next to each line. The ToF is the time between the droplet release and its193

deposit on the canopy. Droplet ToF is shown decreasing with increasing droplet194

size. The 100µm diameter droplets had, on average, 20 times longer ToF than195

400µm diameter droplets. The ToF ratio between the 250µm and the 400µm196

diameter droplets was around 2.197

Figure 4 b shows the evolution of the horizontal median droplet velocity with198

respect to the droplet vertical position. All droplet sizes reached the top canopy199

level at a horizontal velocity approximately equal to the average wind velocity.200

An overshoot of the wind velocity was observed for larger droplets due to their201

inertia, e.g. 250µm droplets are faster than the wind at z≤ 0.2 m.202

3.1.2. Droplet trajectories203

The random wind fluctuations experienced by the droplets lead to a vari-204

ability of trajectories among the simulations. Figure 5 a shows the 5th, 50th205

(median) and 95th percentile of the trajectories of 15 000 droplets under refer-206

ence conditions (cf Table 2). The median is represented by the solid line. The207

5th and 95th percentile are represented by the left and the right dashed line re-208

spectively. The coarser the droplet, the shorter the horizontal distance travelled209

and the dispersion of the travelled distance. The droplets with diameter larger210

13

10-3

10-2

10-1

100

101

Horizontal position [m]

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Vert

ical positio

n [m

]

100 µm

150 µm

200 µm

250 µm

300 µm

400 µm

a) b)

Fig. 5: a) 5th percentile, median and 95th percentile trajectories for 6 different droplet sizes

under standard conditions. b) Deposition pattern of 15 000 droplets with the same size under

standard conditions. The line with the bullets represents the simulated data and the full lines

represents the log-normal fit. The log-normal distribution arithmetic mean and the arithmetic

variance are displayed above each curves. Details on these parameters are available in the

Appendix B.

than 200µm reached the canopy within 1 m from the release position with a211

dispersion shorter than 0.1 m.212

The 95th percentile curve for the 100µm droplet features a plateau between213

0.1 m and 1 m. This plateau arises from a succession of random velocity fluctu-214

ations directed upwards. At a wind speed of 2 m s−1 at 2 m height, the vertical215

velocity fluctuations are equal to u′z = 0.284 ε with ε a random standard Gaus-216

sian value which is in the same range than the settling velocity of droplet of217

100µm (i.e. 0.29 m s−1). Computations (not displayed here for brevity) showed218

that with a higher wind speeds, the plateau forms a bell shape due to the in-219

crease in the strength of the vertical velocity fluctuations.220

The simulated relative deposition patterns over distance is shown in Fig-221

ure 5 b by dashed with bullets. The full line represents the log-normal fit on222

the simulated data. The fitted and simulated data are in good agreement. The223

next subsection assess the effect of the wind speed and the release parameters on224

the arithmetic mean of the log-normal distribution. The value of the arithmetic225

mean has been retrieved with a least square fitting of the log-normal parameters226

on the numerical data using Matlab (MATLAB 9.0, The MathWorks Inc., Nat-227

14

100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Release height [m]

0

1

2

3

4

5

6 E

[m

] 0.2 0.4 0.6 0.8 1

Release height [m]

0

0.2

0.4

0.6

0.8

1

E [

m]

a)

0 1 2 3 4 5 6 7 8

Wind Speed [m s-1]

0

1

2

3

4

5

6

7

8

9

10

E [

m] 0 2 4 6 8

Wind Speed [m s-1

]

0

0.2

0.4

0.6

0.8

E [

m]

b)

5 6 7 8 9 10 11 12 13 14 15

Release velocity [m s-1]

0

0.5

1

1.5

2

2.5

3

E [

m]

c)

0 10 20 30 40 50 60 70 80 90

Release angle β [°]

0

0.5

1

1.5

2

2.5

3

E [

m]

d)

Fig. 6: Effect of the release height, wind speed, release velocity and release angle on the

average of the log-normal fit arithmetic mean E.

ick, MA, USA). More details about the log-normal distribution and the reduced228

parameters are furnished in the Appendix B.229

3.1.3. Average droplet transport230

The results of the arithmetic mean E [m] with respect to variation of the231

ejection height, ejection angle, wind speed and ejection velocity are presented232

in Fig.6. The horizontal distance travelled by a droplet was correlated with233

droplet ToF and wind speed. ToF decreased with decreasing release velocity234

and increasing droplet settling velocity. The release height increased the aver-235

age displacement, mainly for droplet smaller than 250µm. For fine droplets, the236

release height was roughly proportional to the ToF since the droplets quickly237

15

reached their settling velocities, leading to a linear relationship between trav-238

elled distance and release height. For droplets coarser than 200µm, the latter239

relationship was not linear because larger droplets adecelerate during their fall.240

The travelled distance linearly increased with increasing wind speed. Finer241

droplets were more sensitive to wind speed, resulting in steeper slopes in the242

graph of Fig.6 b. Increase in the release velocity slightly decreased the traveled243

distance for the finer droplets (- 20% for 100µm 5-15 m s−1) whilst the decrease244

was substantial for coarse droplets (- 80% for 400µm 5-15 m s−1) which relates245

to droplet inertia.The effect of the release angle β is shown in Fig.6 d. For each246

angle, the average displacement without wind was subtracted to consider the247

effect of these angles. The increase of β leads to a decrease in initial vertical248

velocity and an increase of the initial horizontal velocity, increasing the averaged249

travelled distance. Droplets with diameter ≥ 200µm had an average horizontal250

displacement shorter than 0.5 m for release angle ≤ 60◦.251

3.1.4. Droplet transport of 95th percentile252

X95 represents the downwind distance by which 95 % of the spray volume has253

reached the ground. It corresponds to the final position of the 95th percentile254

trajectories shown in Fig.5 a. This parameter was responsive to the average255

transport of the droplet spray and deposition dispersion. The effect of the main256

parameters on the X95 is shown in Fig.7. The increase of release height was lin-257

ear with increasing X95, similarly to Fig.6 a. However, the decrease of the slope258

with increasing droplet size was stronger than for the average displacement since259

the increase of height also enhanced the deposition variability. The increase of260

wind speed generated a quadratic increase of the X95. This can be explained by261

an increase in the random wind fluctuations which in turn enhanced the vari-262

ability of the droplet trajectories. Therefore, the log-normal curves representing263

the volume distribution over distance were strongly flattened.At windspeeds of264

8 m s−1, more than 5 % of the droplets of 100µm travelled further than 100 m.265

This distance dropped below 1 m for droplets with diameter ≥ 250µm. The266

increase of the release velocity led to a moderate decrease of X95. Thus, acting267

16

100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Release height [m]

0

2

4

6

8

10

12

14

16

18

20

X9

5 [

m]

0.2 0.4 0.6 0.8 1

Release height [m]

0

0.5

1

1.5

2

2.5

3

X9

5 [m

]

a)

0 1 2 3 4 5 6 7 8

Wind Speed [m s-1]

0

20

40

60

80

100

120

X95 [m

]

0 2 4 6 8

Wind Speed [m s-1

]

0

1

2

3

4

5

6

7

X95 [m

]

b)

5 6 7 8 9 10 11 12 13 14 15

Release velocity [m s-1]

0

1

2

3

4

5

6

7

X95 [

m]

c)

0 10 20 30 40 50 60 70 80 90

Release angle β [°]

0

1

2

3

4

5

6

7

X95 [

m]

d)

Fig. 7: Effect of the release height, wind speed, release velocity and release angle on the

distance above which 95 % of the droplets have reach the top canopy level X95.

17

100 150 200 250 300 350 400

Droplet diameter [µm]

0

5

10

15

uz a

t cro

p c

an

op

y [

m s

-1]

Shatter limit

Bounce limit

0.25 m at 5 m s -1

0.25 m at 10 m s -1

0.25 m at 15 m s -1

0.50 m at 5 m s -1

0.50 m at 10 m s -1

0.50 m at 15 m s -1

0.75 m at 5 m s -1

0.75 m at 10 m s -1

0.75 m at 15 m s -1

Settling velocity

a)

100 150 200 250 300 350 400

Droplet diameter [µm]

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

An

gle

with

th

e h

orizo

nta

l d

ire

ctio

n [

°]

0.25 m at 5 m s -1

0.25 m at 10 m s -1

0.25 m at 15 m s -1

0.50 m at 5 m s -1

0.50 m at 10 m s -1

0.50 m at 15 m s -1

0.75 m at 5 m s -1

0.75 m at 10 m s -1

0.75 m at 15 m s -1

b)

Fig. 8: a) Average impact velocity in respect to the droplet diameter for three release heights

and three release velocities. The dashed line shows the velocity above which a droplet would

shatter or bounce while impact a wheat leaf using the adhesion model described in section 2.2.

b) Average droplet trajectory angle at the crop top canopy level with the horizontal direction.

The other simulation parameters were set at standard values (cf Table 2).

solely on the release velocity does not significantly affect drift. The increase of268

β led to an increase of X95, especially for coarser droplets at release angles from269

60◦ to 90◦. At an angle of 60◦, less than 5 % of the droplets with diameter ≥270

200µm were airborne further than 1 m. X95 was strongly influence by droplet271

size due to the higher deposition variability and higher average displacement272

for finer droplets. This means that droplets diameter ≤ 150µm should be min-273

imised within the spray since a significant proportion will travel several metres.274

E and X95 were close to each other for droplets with diameter ≥ 250µm showing275

a low dispersion of droplet trajectories. This low dispersion can be explained276

by their shorter ToF relatively to finer droplets.277

3.1.5. Droplet velocity at top canopy level278

Figure 8 a shows the average droplet vertical velocity at crop height for three279

release heights and three release velocities. The coarser the droplet, the shorter280

is the travelled distance and the faster it may impact on the target. The black281

line shows experimental measurements of settling velocities (Gunn & Kinzer,282

1949). At a release height of 0.5 m, droplets smaller with diameter ≤ 200µm283

reached their settling velocity. At 350µm diameter, the 0.25 m and 10 m s−1284

18

line crosses the 0.5 m 15 m s−1 line showing that the increase of release velocity285

overcomes the increase in flight distance for droplets with higher inertia. The286

black dashed line shows the thresholds for droplet bounce and shatter on a287

wheat leaf predicted for water (Dorr et al., 2015). For the whole range of288

droplet size studied, shatter occurs when the droplets move faster than their289

settling velocity. For standard simulation conditions (i.e. 0.5 m and 10 m s−1)290

droplets larger than 400µm shattered and droplets between 270 and 400µm291

bounced. The mitigation of bounce and shatter can be done by increasing the292

release height or by decreasing the release velocity. Nevertheless, decreasing the293

release velocity was predicted as being less detrimental for the spray drift as294

shown in Fig.6 a,c.295

Droplet trajectory at the top canopy affects the potential droplet reten-296

tion. For graminicide application, vertical trajectories reduce the droplet cap-297

ture probability by the target (Jensen, 2012; Spillman, 1984). Figure 8 b shows298

the average trajectory angles in respect with the horizontal direction under a299

wind of 2 m s−1 at 2 m above the crop. The droplet ToF and the droplet size300

will affect the final horizontal velocity whilst the droplet size, release height301

and the release velocity will determine the final vertical velocity. The fine and302

therefore slow droplets reach the canopy more horizontally than the coarse ones.303

For a release height of 0.5 m, the droplets reached the top of the canopy with304

roughly the same horizontal velocity as shown in Fig.4 b. Therefore, the differ-305

ence in angle between droplet size may be mainly related to the vertical velocity306

component.307

3.1.6. Droplet evaporation308

The effect of the relative humidity Hr, wind speed and droplet size on the309

evaporated fraction is shown in Fig.9. The evaporated fraction was computed310

by subtracting the volume of liquid reaching the top canopy from the initial311

volume released. Evaporation mainly affects droplets with diameter ≤ 150µm.312

Droplets with diameter ≥ 250µm had moderate evaporation, i.e. ≤ 3% for the313

worst scenario. Therefore, for droplet ≥ 250µm diameter, the evaporation may314

19

100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm

40 45 50 55 60 65 70 75 80

HR [%]

0

10

20

30

40

50

60

70

80

90

100

Vo

lum

e e

vap

ora

te [

%]

a)

0 1 2 3 4 5 6 7 8

Wind Speed [m s -1 ]

0

10

20

30

40

50

60

70

80

90

100

Vo

lum

e e

va

po

rate

[%

]

b)

Fig. 9: Evolution of the relative volume evaporate in respect to the relative humidity and the

wind speed.

not be a concern. The evaporation model does not take into account the small315

increase in vapour pressure in the surrounding air due to the droplet evaporation.316

Therefore the evaporation rate observed in real conditions could be lower.317

3.2. Polydisperse sprays318

3.2.1. Deposition319

Figure 10 a shows the volume of spray airborne with respect to the distance320

from the nozzle for twelve simulations with different Dv50 and RSF . As ex-321

pected, increasing Dv50 reduces the volume of airborne spray. Increasing Dv50322

from 150µm to 350µm reduces the airborne spray at 2 m from 20 % to 2 %.323

Comparison of sprays with the same Dv50 shows that lower the RSF can re-324

duce drift. For a Dv50 of 150µm, decreasing of RSF from 1 to 0.6 produces a325

reduction from 20 % to 12 % of the airborne spray at 2 m. Table 3 summarises326

the airborne spray reduction at several distances induced by reducing the RSF327

from 1.0 to 0.6 computed as: 100Drift 0.6

Drift 1.0. The drift reduction produced by328

the RSF reduction increased with the Dv50 because the coarser the spray, the329

greater the relative reduction of the fine droplets. For the spray with a Dv50330

of 250µm, drift reduction was around 80 % which corresponds to a three star331

rating in the LERAP scheme (Butler Ellis, Alanis, Lane, Tuck, Nuyttens &332

20

Table 3: Airborne spray reduction [%] induced by a RSF reduction from 1.0 to 0.6. The

airborne spray reduction is given for each Dv50 at 5 distances from the release point.

Distance [m]

2 4 6 8 10

Dv50

[µm

]

150 43.4 56.0 55.7 49.2 42.9

200 67.3 76.1 76.0 73.5 70.8

225 74.2 81.3 81.2 79.9 77.2

250 80.3 85.4 85.4 83.5 80.4

300 87.0 90.6 90.0 88.3 87.0

350 90.6 93.1 93.4 93.1 92.7

van de Zande, 2017). For each Dv50 the drift reduction appeared to be roughly333

constant over distance.334

3.2.2. Retention335

Figure 11 shows the relative volume of each droplet impact for the twelve336

simulated sprays on a wheat leaf. For a given RSF , the increase of Dv50 leads337

to a monotonic decrease of adhesion and the emergence of bounce and shatter338

due to a progressive increase of larger droplet proportion. Reduction of RSF339

enhanced one outcome according to the Dv50, for Dv50 ≤ 250µm there was an340

increase of the adhesion whilst bounce increased for Dv50 ≥ 300µm. For stan-341

dard conditions, the diameter threshold between adhesion and bounce is around342

270µm as shown in Fig.8 a. Therefore, a RSF reduction may be detrimental343

if the Dv50 is not in the adequate range as has already been noted in previous344

theoretical work (Massinon, De Cock, Ouled Taleb Salah & Lebeau, 2016).345

4. Discussion346

The study of the droplet transport dynamic has shown that the size of a347

droplet affects its trajectory. The finer the droplet longer its time in the air348

making it more sensitive to evaporation and drift. The droplet ToF can be349

21

0 2 4 6 8 10 12 14 16 18 20

Distance [m]

10-4

10-3

10-2

10-1

100

101

102

Spra

y a

irbone [%

]

Span = 0.6 VMD = 150 µm

Span = 0.6 VMD = 200 µm

Span = 0.6 VMD = 225 µm

Span = 0.6 VMD = 250 µm

Span = 0.6 VMD = 300 µm

Span = 0.6 VMD = 350 µm

Span = 1.0 VMD = 150 µm

Span = 1.0 VMD = 200 µm

Span = 1.0 VMD = 225 µm

Span = 1.0 VMD = 250 µm

Span = 1.0 VMD = 300 µm

Span = 1.0 VMD = 350 µm

Fig. 10: Volume of airborne spray in respect with the distance. Twelve sprays were simulated

with different Dv50 and RSF values. The outcomes have been determined at the top canopy

level using the models described in the section 2.2 for a release height of 0.5 m and a release

speed of 10 m s−1.

22

0

0.2

0.4

0.6

0.8

1

Rela

tive v

olu

me

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Rela

tive v

olu

me

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

150 200 225 250 300 350

0.6

1.0

Dv50 [µm]

RS

F [-]

A SB A SB A SB A SB A SBA SB

A SB A SB A SB A SB A SBA SB

Fig. 11: Droplet impact outcome predictions at top canopy level expressed in relative volume.

A, B and S correspond to adhesion, bounce and shatter respectively.

shortened by decreasing the release height or by increasing the release veloc-350

ity. Release angle and release velocity have moderate effects on fine droplets.351

However for coarse droplets, increasing the release velocity increases the droplet352

velocity at the canopy and thus its outcome during impaction. The change of353

the release angle from vertical to horizontal direction leads to an increase in354

the travelled distance arising from both the initial horizontal velocity and the355

increase in ToF. An optimum value may be around 60◦. The wind speed is356

enhances the average droplet travelled distance linearly and the maximum dis-357

tance quadratically. However, with droplets of 250µm, and 8 m s−1 wind speed358

95 % of the spray reached canopy top level below 1 m from the release point. The359

shatter threshold on a wheat leaf was reached by droplets larger than 400µm360

when the release height is at 0.5 m and the release speed at 10 m s−1. With361

tthese initial conditions, bounce occurs for droplet between 270 and 400µm.362

Therefore, droplets with a diameter from 200 to 270µm have a low drift poten-363

tial and may not shatter or bounce on a wheat leaf.364

23

For a polydisperse sprays, the overall behaviour can be seen as the combi-365

nation of drop size distribution and the properties of each droplet size. Drift366

and the volume of droplet adhesion decrease with increasing Dv50. Narrowing367

the RSF of the spray may solve this problem. A spray with a Dv50 of 225µm368

and a RSF of 0.6 released at 0.5 m at 10 m s−1 above the crop produces low369

drift with moderate kinetic energy at the crop canopy level. Using a Weibull370

distribution, this spray would have a Dv10 of 152µm, Dv90 of 288µm with 1.4 %371

of the droplet volume ≤ 100µm diameter and 9.5 % ≤ 150µm diameter. The372

narrowing of the spray drift may be detrimental when the Dv50 is too small or373

too large which would enhanced drift or decreases retention.374

5. Conclusion375

A combined Lagrangian droplet transport and retention models has been376

presented. The deposition over distance had a log-normal distribution with a377

dispersion and average distance larger for finer droplets. The results of numerical378

simulations showed that droplets with diameters ranging between 200µm and379

250µm offered high control of deposition by combining a low drift potential and380

moderate kinetic energy at the top of the canopy. The reduction of the RSF381

from 1.0 to 0.6 is an effective way to mitigate deposition and retention losses.382

A fourfold reduction of the drift volume at a distance of 2 m from the nozzle383

was observed for a spray with a Dv50 = 225µm when the RSF was reduced384

from 1.0 to 0.6. Under this scenario, an increase in the volumetric proportion385

of adhering droplets on a wheat leaf from 63 to 78% was shown. Therefore,386

strategies to control the droplet size distribution in terms of Dv50 and RSF387

may offer promising solutions for reducing adverse impacts on environment of388

spray applications.389

Further work should be carried out on the experimental assessment of the390

performance of such sprays in term of drift reduction and retention on target.391

Sprays with a RSF around 0.6 and a Dv50 of 225µm appear to feasible using392

rotary atomisers (Qi et al., 2008).393

24

Acknowledgments394

This work was supported by the Fonds De La Recherche Scientifique - FNRS,395

under the FRIA grant n◦ 97 364.396

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29

Appendix A527

The mass flux is given by:528

m = AShgDg

dρg(Yv,s − Ys,∞) (A.1)

with A the droplet area [m2], d the droplet diameter, Shg the gaseous Sher-529

wood number [-], Dg the molecular diffusion [m s−2], Yv,s and Yv,∞ are the vapor530

mass fractions at the droplet interface and far from the droplet respectively [-].531

For a diameters less than 5 mm, Shg can be computed as:532

Shg = 1.61 + 0.718Re0.5Sc0.33g (A.2)

with Scg the Schmidt number for the gaseous phase [-] expressed by: Scg =533

νDg

.534

The molar fraction at the droplet surface Yv,s is computed as:535

Yv,s = yvlMl

Mt(A.3)

with yvl = PsatPtot

and Mt = yvlMl + (1 − yvl )Mg. Ml and Mg are the molar536

mass of the liquid and the gaseous phase respectively [g mol−1]. Therefore, at537

atmospheric pressure the vapour mass fraction at the droplet interface neigh-538

bourhood, Yv,s, reads:539

Yv,s =Psat

Psat + (Ptot − Psat) Mg

Ml

(A.4)

The vapour pressure in the far field is computed using the relative humidity540

Hr:541

Yinf = HrPsat

HrPsat + (Ptot −HrPsat)(Mg

Ml

) (A.5)

The droplet exchanges heat with the air by convection. The heat flux be-542

tween the droplet surface and the surrounding air Qd [J s−1] reads:543

Qd = SlNugλgd

(Tinf − Tl) (A.6)

30

with Tl, Tinf the temperature of the droplet and far from the droplet inter-544

face respectively [K], λg the thermal conductivity of the gaseous phase [W m−1 K−1].545

The gaseous Nusselt number Nug [-] is a function of the gaseous Reynolds and546

the gaseous Prandlt number Prg [-]:547

Nug = 1.61 + 0.718√Re 3√Prg (A.7)

with Prg =Cpµgλg

548

The thermal balance is given by difference between the convection heat flux549

and the latent heat flux:550

Ql = Qd − mLv (A.8)

with Lv the vaporisation latent heat of water [J kg−1] can be expressed as a551

function of the reduced temperature Tr:552

Lv = 52.05310e6 (1− Tr)0.3199−0.212Tr+0.25795T 2r (A.9)

with Tr = Tl+273Tc

for water Tc=647.13 K.553

The temperature at each time step is retrieved by integrate:

VlρlCp∂dTldt

= SlNugλgd

(Tinf − Tl)− Lvm (A.10)

Cp the heat capacity [J K−1]. For water, the heat capacity is given by:554

Cp = 276730− 2090.1T + 8.125T 2 − 0.014116T 3 + 9.3701e6T 4 (A.11)

Appendix B555

The cumulative distribution function (CDF) of a log-normal distribution is

defined as:

CDF =1

xσn√

2πe− (ln(x)−µn)2

2σ2n (B.1)

31

with σn and µn the two log-normal distribution parameters. From these param-556

eters reduced variables can be extracted: the arithmetic mean E = eµn+0.5σ2n557

and the arithmetic variance Var = e2µn+2σ2n

(eσ

2n − 1

).558

32